MULTILEVEL IMAGE-RECONSTRUCTION WITH NATURAL PIXELS

The sampled Radon transform of a two-dimensional (2D) function can be represented as a continuous linear map A : L(2)(Omega) --> R(N), where (Au)(j) = (u, psi(j)) and psi(j) is the characteristic function of a strip through Omega approximating the set of line integrals in the sam ple. The image reconstruction problem is: given a vector b is an eleme nt of R(N), find an image (or density function) u(x, y) such that Au = b. In general there are infinitely many solutions; we seek the soluti on with minimal 2-norm, which leads to a matrix equation Bw = b, where B is a square dense matrix with several convenient properties. We ana lyze the use of Gauss-Seidel iteration applied to the problem, observi ng that while the iteration formally converges, there exists a near nu ll space into which the error vectors migrate, after which the iterati on stalls. The null space and near null space of B are characterized i n order to develop a multilevel scheme. Based on the principles of the multilevel projection method (PML), this scheme leads to somewhat imp roved performance. Its primary utility, however, is that it facilitate s the development of a PML-based method for spotlight tomography, that is, local grid refinement over a portion of the image in which featur es of interest can be resolved at finer scale than is possible globall y.